L(s) = 1 | + (0.707 − 0.707i)3-s + (−2.18 − 0.466i)5-s + 1.00·7-s − 1.00i·9-s + (1.89 − 1.89i)11-s + (−2.65 + 2.65i)13-s + (−1.87 + 1.21i)15-s + 1.73i·17-s + (−5.33 − 5.33i)19-s + (0.707 − 0.707i)21-s + 0.160·23-s + (4.56 + 2.04i)25-s + (−0.707 − 0.707i)27-s + (−2.70 − 2.70i)29-s − 4.64·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (−0.977 − 0.208i)5-s + 0.378·7-s − 0.333i·9-s + (0.571 − 0.571i)11-s + (−0.737 + 0.737i)13-s + (−0.484 + 0.314i)15-s + 0.421i·17-s + (−1.22 − 1.22i)19-s + (0.154 − 0.154i)21-s + 0.0334·23-s + (0.912 + 0.408i)25-s + (−0.136 − 0.136i)27-s + (−0.501 − 0.501i)29-s − 0.833·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5570786002\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5570786002\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (2.18 + 0.466i)T \) |
good | 7 | \( 1 - 1.00T + 7T^{2} \) |
| 11 | \( 1 + (-1.89 + 1.89i)T - 11iT^{2} \) |
| 13 | \( 1 + (2.65 - 2.65i)T - 13iT^{2} \) |
| 17 | \( 1 - 1.73iT - 17T^{2} \) |
| 19 | \( 1 + (5.33 + 5.33i)T + 19iT^{2} \) |
| 23 | \( 1 - 0.160T + 23T^{2} \) |
| 29 | \( 1 + (2.70 + 2.70i)T + 29iT^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 + (5.35 + 5.35i)T + 37iT^{2} \) |
| 41 | \( 1 + 9.89iT - 41T^{2} \) |
| 43 | \( 1 + (-7.23 - 7.23i)T + 43iT^{2} \) |
| 47 | \( 1 - 4.79iT - 47T^{2} \) |
| 53 | \( 1 + (3.44 + 3.44i)T + 53iT^{2} \) |
| 59 | \( 1 + (0.101 - 0.101i)T - 59iT^{2} \) |
| 61 | \( 1 + (-6.01 - 6.01i)T + 61iT^{2} \) |
| 67 | \( 1 + (9.04 - 9.04i)T - 67iT^{2} \) |
| 71 | \( 1 + 4.60iT - 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 5.73T + 79T^{2} \) |
| 83 | \( 1 + (2.04 - 2.04i)T - 83iT^{2} \) |
| 89 | \( 1 + 15.0iT - 89T^{2} \) |
| 97 | \( 1 - 3.84iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.960202707103179772626674940604, −8.035701743845722329776012563269, −7.28671924526851862126325231660, −6.71187383834937459528168781078, −5.62821255307354177348166544586, −4.44734428029462189086268915735, −3.96615113959189424102633958809, −2.77876507327635998560641416645, −1.67703322658443138019141767348, −0.18484625272622658842867940980,
1.71645510552683729856465491103, 2.94344277362157242770517901254, 3.83181136891827980320520767806, 4.53825514149869977578946639047, 5.37983445019070052733403111968, 6.59365230001185303737964502936, 7.40764926871297086841389637477, 8.022550622913724304329604589969, 8.692001235442617923421762192587, 9.575535579029906174241039584379